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Theorem metcnpi 21172
Description: Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 21169. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
Hypotheses
Ref Expression
metcn.2  |-  J  =  ( MetOpen `  C )
metcn.4  |-  K  =  ( MetOpen `  D )
Assertion
Ref Expression
metcnpi  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
A ) )
Distinct variable groups:    x, y, F    x, J, y    x, K, y    x, X, y   
x, Y, y    x, A, y    x, C, y   
x, D, y    x, P, y

Proof of Theorem metcnpi
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  F  e.  ( ( J  CnP  K ) `  P ) )
2 simpll 753 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  C  e.  ( *Met `  X
) )
3 simplr 755 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  D  e.  ( *Met `  Y
) )
4 eqid 2457 . . . . . . . . 9  |-  U. J  =  U. J
54cnprcl 19872 . . . . . . . 8  |-  ( F  e.  ( ( J  CnP  K ) `  P )  ->  P  e.  U. J )
65adantl 466 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  P  e.  U. J )
7 metcn.2 . . . . . . . . 9  |-  J  =  ( MetOpen `  C )
87mopnuni 21069 . . . . . . . 8  |-  ( C  e.  ( *Met `  X )  ->  X  =  U. J )
98ad2antrr 725 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  X  =  U. J )
106, 9eleqtrrd 2548 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  P  e.  X )
11 metcn.4 . . . . . . 7  |-  K  =  ( MetOpen `  D )
127, 11metcnp 21169 . . . . . 6  |-  ( ( C  e.  ( *Met `  X )  /\  D  e.  ( *Met `  Y
)  /\  P  e.  X )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <->  ( F : X --> Y  /\  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
z ) ) ) )
132, 3, 10, 12syl3anc 1228 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  ( F  e.  ( ( J  CnP  K ) `  P )  <-> 
( F : X --> Y  /\  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  <  x  -> 
( ( F `  P ) D ( F `  y ) )  <  z ) ) ) )
141, 13mpbid 210 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  ( F : X --> Y  /\  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
z ) ) )
1514simprd 463 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  <  x  ->  ( ( F `  P ) D ( F `  y ) )  <  z ) )
16 breq2 4460 . . . . . 6  |-  ( z  =  A  ->  (
( ( F `  P ) D ( F `  y ) )  <  z  <->  ( ( F `  P ) D ( F `  y ) )  < 
A ) )
1716imbi2d 316 . . . . 5  |-  ( z  =  A  ->  (
( ( P C y )  <  x  ->  ( ( F `  P ) D ( F `  y ) )  <  z )  <-> 
( ( P C y )  <  x  ->  ( ( F `  P ) D ( F `  y ) )  <  A ) ) )
1817rexralbidv 2976 . . . 4  |-  ( z  =  A  ->  ( E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
z )  <->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
A ) ) )
1918rspccv 3207 . . 3  |-  ( A. z  e.  RR+  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
z )  ->  ( A  e.  RR+  ->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
A ) ) )
2015, 19syl 16 . 2  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  F  e.  ( ( J  CnP  K ) `  P ) )  ->  ( A  e.  RR+  ->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
A ) ) )
2120impr 619 1  |-  ( ( ( C  e.  ( *Met `  X
)  /\  D  e.  ( *Met `  Y
) )  /\  ( F  e.  ( ( J  CnP  K ) `  P )  /\  A  e.  RR+ ) )  ->  E. x  e.  RR+  A. y  e.  X  ( ( P C y )  < 
x  ->  ( ( F `  P ) D ( F `  y ) )  < 
A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   U.cuni 4251   class class class wbr 4456   -->wf 5590   ` cfv 5594  (class class class)co 6296    < clt 9645   RR+crp 11245   *Metcxmt 18529   MetOpencmopn 18534    CnP ccnp 19852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-topgen 14860  df-psmet 18537  df-xmet 18538  df-bl 18540  df-mopn 18541  df-top 19525  df-bases 19527  df-topon 19528  df-cnp 19855
This theorem is referenced by: (None)
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